MCA, Vol. 30, Pages 131: The Algebraic Theory of Operator Matrix Polynomials with Applications to Aeroelasticity in Flight Dynamics and Control

Mathematical and Computational Applications doi: 10.3390/mca30060131

Authors:
Belkacem Bekhiti
Kamel Hariche
Vasilii Zaitsev
Guangren R. Duan
Abdel-Nasser Sharkawy

This paper develops an algebraic framework for operator matrix polynomials and demonstrates its application to control-design problems in aeroservoelastic systems. We present constructive spectral-factorization and linearization tools (block spectral divisors, companion forms and realization algorithms) that enable systematic block-pole assignment for large-scale MIMO models. Building on this theory, an adaptive block-pole placement strategy is proposed and cast in a practical implementation that augments a nominal state-feedback law with a compact neural-network compensator (single hidden layer) to handle un-modeled nonlinearities and uncertainty. The method requires state feedback and the system’s nominal model and admits Laplace-domain analysis and straightforward implementation for a two-degree-of-freedom aeroelastic wing with cubic stiffness nonlinearity and Roger aerodynamic lag is validated in MATLAB R2023a. Comprehensive simulations (Runge–Kutta 4) for different excitations and step disturbances demonstrate the approach’s advantages: compared with Eigenstructure assignment, LQR and H2-control, the proposed method achieves markedly better robustness and transient performance (e.g., closed-loop ‖H(iω)‖2 ≈ 4.64, condition number χ ≈ 11.19, and reduced control efforts μ ≈ 0.41, while delivering faster transients and tighter regulation (rise time ≈ 0.35 s, settling time ≈ 1.10 s, overshoot ≈ 6.2%, steady-state error ≈ 0.9%, disturbance-rejection ≈ 92%). These results confirm that algebraic operator-polynomial techniques, combined with a compact adaptive NN augmentation, provide a well-conditioned, low-effort solution for robust control of aeroelastic systems.

Source link

Belkacem Bekhiti www.mdpi.com